




import heapq




def 		lowerCAmelCase_ (						lowerCamelCase ):
				__magic_name__						:						list[list]										=[]

				# for each node and his adjacency list add them and the rank of the node to queue
				# using heapq module the queue will be filled like a Priority Queue
				# heapq works with a min priority queue, so I used -1*len(v) to build it
				for key, value in graph.items():
								# O(log(n))
								heapq.heappush(lowerCamelCase						,  [-1 * len(lowerCamelCase ), (key, value)] )

				# chosen_vertices = set of chosen vertices
				__magic_name__						:						Tuple										=set()

				# while queue isn't empty and there are still edges
				#   (queue[0][0] is the rank of the node with max rank)
				while queue and queue[0][0] != 0:
								# extract vertex with max rank from queue and add it to chosen_vertices
								__magic_name__						:						Tuple										=heapq.heappop(lowerCamelCase )[1][0]
								chosen_vertices.add(lowerCamelCase )

								# Remove all arcs adjacent to argmax
								for elem in queue:
												# if v haven't adjacent node, skip
												if elem[0] == 0:
																continue
												# if argmax is reachable from elem
												# remove argmax from elem's adjacent list and update his rank
												if argmax in elem[1][1]:
																__magic_name__						:						Tuple										=elem[1][1].index(lowerCamelCase )
																del elem[1][1][index]
																elem[0] += 1
        # re-order the queue
								heapq.heapify(lowerCamelCase )
				return chosen_vertices


if __name__ == "__main__":
							import doctest

							doctest.testmod()

							UpperCAmelCase_		:					Optional[int]          =  {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
							print(F"""Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}""")









def  lowerCAmelCase_	(lowerCAmelCase__:		int  ):





			"""simple docstring"""


			if num <= 0:
						raise ValueError("""Input must be a positive integer"""  )

			UpperCAmelCase_:		Tuple							      =     [True] * (num + 1)

			UpperCAmelCase_:		Tuple							      =     2
			while p * p <= num:
						if primes[p]:
									for i in range(p * p		,						num + 1		,						lowerCAmelCase__  ):
												UpperCAmelCase_:		Optional[Any]							      =     False
						p += 1

			return [prime for prime in range(2		,						num + 1  ) if primes[prime]]


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	a      :  Optional[Any]			     =			int(input('Enter a positive integer: ').strip())
	print(prime_sieve_eratosthenes(user_num))






